Theorem xpsfrnel2 16225

Description: Elementhood in the target space of the function 𝐹 appearing in xpsval 16232. (Contributed by Mario Carneiro, 15-Aug-2015.)

Assertion

Ref	Expression
xpsfrnel2	⊢ (◡({𝑋} +𝑐 {𝑌}) ∈ X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵))

Distinct variable groups:   𝐴,𝑘   𝐵,𝑘   𝑘,𝑋   𝑘,𝑌

Proof of Theorem xpsfrnel2

Step	Hyp	Ref	Expression
1		xpsfrnel 16223	. 2 ⊢ (◡({𝑋} +𝑐 {𝑌}) ∈ X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) ↔ (◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 ∧ (◡({𝑋} +𝑐 {𝑌})‘∅) ∈ 𝐴 ∧ (◡({𝑋} +𝑐 {𝑌})‘1𝑜) ∈ 𝐵))
2		0ex 4790	. . . . . . . . . 10 ⊢ ∅ ∈ V
3	2	prid1 4297	. . . . . . . . 9 ⊢ ∅ ∈ {∅, 1𝑜}
4		df2o3 7573	. . . . . . . . 9 ⊢ 2𝑜 = {∅, 1𝑜}
5	3, 4	eleqtrri 2700	. . . . . . . 8 ⊢ ∅ ∈ 2𝑜
6		fndm 5990	. . . . . . . 8 ⊢ (◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 → dom ◡({𝑋} +𝑐 {𝑌}) = 2𝑜)
7	5, 6	syl5eleqr 2708	. . . . . . 7 ⊢ (◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 → ∅ ∈ dom ◡({𝑋} +𝑐 {𝑌}))
8		xpsc 16217	. . . . . . . . 9 ⊢ ◡({𝑋} +𝑐 {𝑌}) = (({∅} × {𝑋}) ∪ ({1𝑜} × {𝑌}))
9	8	dmeqi 5325	. . . . . . . 8 ⊢ dom ◡({𝑋} +𝑐 {𝑌}) = dom (({∅} × {𝑋}) ∪ ({1𝑜} × {𝑌}))
10		dmun 5331	. . . . . . . 8 ⊢ dom (({∅} × {𝑋}) ∪ ({1𝑜} × {𝑌})) = (dom ({∅} × {𝑋}) ∪ dom ({1𝑜} × {𝑌}))
11	9, 10	eqtri 2644	. . . . . . 7 ⊢ dom ◡({𝑋} +𝑐 {𝑌}) = (dom ({∅} × {𝑋}) ∪ dom ({1𝑜} × {𝑌}))
12	7, 11	syl6eleq 2711	. . . . . 6 ⊢ (◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 → ∅ ∈ (dom ({∅} × {𝑋}) ∪ dom ({1𝑜} × {𝑌})))
13		elun 3753	. . . . . . 7 ⊢ (∅ ∈ (dom ({∅} × {𝑋}) ∪ dom ({1𝑜} × {𝑌})) ↔ (∅ ∈ dom ({∅} × {𝑋}) ∨ ∅ ∈ dom ({1𝑜} × {𝑌})))
14	2	eldm 5321	. . . . . . . . 9 ⊢ (∅ ∈ dom ({∅} × {𝑋}) ↔ ∃𝑘∅({∅} × {𝑋})𝑘)
15		brxp 5147	. . . . . . . . . . 11 ⊢ (∅({∅} × {𝑋})𝑘 ↔ (∅ ∈ {∅} ∧ 𝑘 ∈ {𝑋}))
16		elsni 4194	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ {𝑋} → 𝑘 = 𝑋)
17		vex 3203	. . . . . . . . . . . . 13 ⊢ 𝑘 ∈ V
18	16, 17	syl6eqelr 2710	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ {𝑋} → 𝑋 ∈ V)
19	18	adantl 482	. . . . . . . . . . 11 ⊢ ((∅ ∈ {∅} ∧ 𝑘 ∈ {𝑋}) → 𝑋 ∈ V)
20	15, 19	sylbi 207	. . . . . . . . . 10 ⊢ (∅({∅} × {𝑋})𝑘 → 𝑋 ∈ V)
21	20	exlimiv 1858	. . . . . . . . 9 ⊢ (∃𝑘∅({∅} × {𝑋})𝑘 → 𝑋 ∈ V)
22	14, 21	sylbi 207	. . . . . . . 8 ⊢ (∅ ∈ dom ({∅} × {𝑋}) → 𝑋 ∈ V)
23		dmxpss 5565	. . . . . . . . . 10 ⊢ dom ({1𝑜} × {𝑌}) ⊆ {1𝑜}
24	23	sseli 3599	. . . . . . . . 9 ⊢ (∅ ∈ dom ({1𝑜} × {𝑌}) → ∅ ∈ {1𝑜})
25		elsni 4194	. . . . . . . . 9 ⊢ (∅ ∈ {1𝑜} → ∅ = 1𝑜)
26		1n0 7575	. . . . . . . . . . . 12 ⊢ 1𝑜 ≠ ∅
27	26	neii 2796	. . . . . . . . . . 11 ⊢ ¬ 1𝑜 = ∅
28	27	pm2.21i 116	. . . . . . . . . 10 ⊢ (1𝑜 = ∅ → 𝑋 ∈ V)
29	28	eqcoms 2630	. . . . . . . . 9 ⊢ (∅ = 1𝑜 → 𝑋 ∈ V)
30	24, 25, 29	3syl 18	. . . . . . . 8 ⊢ (∅ ∈ dom ({1𝑜} × {𝑌}) → 𝑋 ∈ V)
31	22, 30	jaoi 394	. . . . . . 7 ⊢ ((∅ ∈ dom ({∅} × {𝑋}) ∨ ∅ ∈ dom ({1𝑜} × {𝑌})) → 𝑋 ∈ V)
32	13, 31	sylbi 207	. . . . . 6 ⊢ (∅ ∈ (dom ({∅} × {𝑋}) ∪ dom ({1𝑜} × {𝑌})) → 𝑋 ∈ V)
33	12, 32	syl 17	. . . . 5 ⊢ (◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 → 𝑋 ∈ V)
34		1on 7567	. . . . . . . . . . 11 ⊢ 1𝑜 ∈ On
35	34	elexi 3213	. . . . . . . . . 10 ⊢ 1𝑜 ∈ V
36	35	prid2 4298	. . . . . . . . 9 ⊢ 1𝑜 ∈ {∅, 1𝑜}
37	36, 4	eleqtrri 2700	. . . . . . . 8 ⊢ 1𝑜 ∈ 2𝑜
38	37, 6	syl5eleqr 2708	. . . . . . 7 ⊢ (◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 → 1𝑜 ∈ dom ◡({𝑋} +𝑐 {𝑌}))
39	38, 11	syl6eleq 2711	. . . . . 6 ⊢ (◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 → 1𝑜 ∈ (dom ({∅} × {𝑋}) ∪ dom ({1𝑜} × {𝑌})))
40		elun 3753	. . . . . . 7 ⊢ (1𝑜 ∈ (dom ({∅} × {𝑋}) ∪ dom ({1𝑜} × {𝑌})) ↔ (1𝑜 ∈ dom ({∅} × {𝑋}) ∨ 1𝑜 ∈ dom ({1𝑜} × {𝑌})))
41		dmxpss 5565	. . . . . . . . . 10 ⊢ dom ({∅} × {𝑋}) ⊆ {∅}
42	41	sseli 3599	. . . . . . . . 9 ⊢ (1𝑜 ∈ dom ({∅} × {𝑋}) → 1𝑜 ∈ {∅})
43		elsni 4194	. . . . . . . . 9 ⊢ (1𝑜 ∈ {∅} → 1𝑜 = ∅)
44	27	pm2.21i 116	. . . . . . . . 9 ⊢ (1𝑜 = ∅ → 𝑌 ∈ V)
45	42, 43, 44	3syl 18	. . . . . . . 8 ⊢ (1𝑜 ∈ dom ({∅} × {𝑋}) → 𝑌 ∈ V)
46	35	eldm 5321	. . . . . . . . 9 ⊢ (1𝑜 ∈ dom ({1𝑜} × {𝑌}) ↔ ∃𝑘1𝑜({1𝑜} × {𝑌})𝑘)
47		brxp 5147	. . . . . . . . . . 11 ⊢ (1𝑜({1𝑜} × {𝑌})𝑘 ↔ (1𝑜 ∈ {1𝑜} ∧ 𝑘 ∈ {𝑌}))
48		elsni 4194	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ {𝑌} → 𝑘 = 𝑌)
49	48, 17	syl6eqelr 2710	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ {𝑌} → 𝑌 ∈ V)
50	49	adantl 482	. . . . . . . . . . 11 ⊢ ((1𝑜 ∈ {1𝑜} ∧ 𝑘 ∈ {𝑌}) → 𝑌 ∈ V)
51	47, 50	sylbi 207	. . . . . . . . . 10 ⊢ (1𝑜({1𝑜} × {𝑌})𝑘 → 𝑌 ∈ V)
52	51	exlimiv 1858	. . . . . . . . 9 ⊢ (∃𝑘1𝑜({1𝑜} × {𝑌})𝑘 → 𝑌 ∈ V)
53	46, 52	sylbi 207	. . . . . . . 8 ⊢ (1𝑜 ∈ dom ({1𝑜} × {𝑌}) → 𝑌 ∈ V)
54	45, 53	jaoi 394	. . . . . . 7 ⊢ ((1𝑜 ∈ dom ({∅} × {𝑋}) ∨ 1𝑜 ∈ dom ({1𝑜} × {𝑌})) → 𝑌 ∈ V)
55	40, 54	sylbi 207	. . . . . 6 ⊢ (1𝑜 ∈ (dom ({∅} × {𝑋}) ∪ dom ({1𝑜} × {𝑌})) → 𝑌 ∈ V)
56	39, 55	syl 17	. . . . 5 ⊢ (◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 → 𝑌 ∈ V)
57	33, 56	jca 554	. . . 4 ⊢ (◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 → (𝑋 ∈ V ∧ 𝑌 ∈ V))
58	57	3ad2ant1 1082	. . 3 ⊢ ((◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 ∧ (◡({𝑋} +𝑐 {𝑌})‘∅) ∈ 𝐴 ∧ (◡({𝑋} +𝑐 {𝑌})‘1𝑜) ∈ 𝐵) → (𝑋 ∈ V ∧ 𝑌 ∈ V))
59		elex 3212	. . . 4 ⊢ (𝑋 ∈ 𝐴 → 𝑋 ∈ V)
60		elex 3212	. . . 4 ⊢ (𝑌 ∈ 𝐵 → 𝑌 ∈ V)
61	59, 60	anim12i 590	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∈ V ∧ 𝑌 ∈ V))
62		3anass 1042	. . . 4 ⊢ ((◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 ∧ (◡({𝑋} +𝑐 {𝑌})‘∅) ∈ 𝐴 ∧ (◡({𝑋} +𝑐 {𝑌})‘1𝑜) ∈ 𝐵) ↔ (◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 ∧ ((◡({𝑋} +𝑐 {𝑌})‘∅) ∈ 𝐴 ∧ (◡({𝑋} +𝑐 {𝑌})‘1𝑜) ∈ 𝐵)))
63		xpscfn 16219	. . . . . 6 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → ◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜)
64	63	biantrurd 529	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (((◡({𝑋} +𝑐 {𝑌})‘∅) ∈ 𝐴 ∧ (◡({𝑋} +𝑐 {𝑌})‘1𝑜) ∈ 𝐵) ↔ (◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 ∧ ((◡({𝑋} +𝑐 {𝑌})‘∅) ∈ 𝐴 ∧ (◡({𝑋} +𝑐 {𝑌})‘1𝑜) ∈ 𝐵))))
65		xpsc0 16220	. . . . . . 7 ⊢ (𝑋 ∈ V → (◡({𝑋} +𝑐 {𝑌})‘∅) = 𝑋)
66	65	eleq1d 2686	. . . . . 6 ⊢ (𝑋 ∈ V → ((◡({𝑋} +𝑐 {𝑌})‘∅) ∈ 𝐴 ↔ 𝑋 ∈ 𝐴))
67		xpsc1 16221	. . . . . . 7 ⊢ (𝑌 ∈ V → (◡({𝑋} +𝑐 {𝑌})‘1𝑜) = 𝑌)
68	67	eleq1d 2686	. . . . . 6 ⊢ (𝑌 ∈ V → ((◡({𝑋} +𝑐 {𝑌})‘1𝑜) ∈ 𝐵 ↔ 𝑌 ∈ 𝐵))
69	66, 68	bi2anan9 917	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (((◡({𝑋} +𝑐 {𝑌})‘∅) ∈ 𝐴 ∧ (◡({𝑋} +𝑐 {𝑌})‘1𝑜) ∈ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)))
70	64, 69	bitr3d 270	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → ((◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 ∧ ((◡({𝑋} +𝑐 {𝑌})‘∅) ∈ 𝐴 ∧ (◡({𝑋} +𝑐 {𝑌})‘1𝑜) ∈ 𝐵)) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)))
71	62, 70	syl5bb 272	. . 3 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → ((◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 ∧ (◡({𝑋} +𝑐 {𝑌})‘∅) ∈ 𝐴 ∧ (◡({𝑋} +𝑐 {𝑌})‘1𝑜) ∈ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)))
72	58, 61, 71	pm5.21nii 368	. 2 ⊢ ((◡({𝑋} +𝑐 {𝑌}) Fn 2𝑜 ∧ (◡({𝑋} +𝑐 {𝑌})‘∅) ∈ 𝐴 ∧ (◡({𝑋} +𝑐 {𝑌})‘1𝑜) ∈ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵))
73	1, 72	bitri 264	1 ⊢ (◡({𝑋} +𝑐 {𝑌}) ∈ X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵))

Syntax hints:   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990  Vcvv 3200   ∪ cun 3572  ∅c0 3915  ifcif 4086  {csn 4177  {cpr 4179   class class class wbr 4653   × cxp 5112  ^◡ccnv 5113  dom cdm 5114  Oncon0 5723   Fn wfn 5883  ‘cfv 5888  (class class class)co 6650  1_𝑜c1o 7553  2_𝑜c2o 7554  Xcixp 7908   +_𝑐 ccda 8989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-cda 8990

This theorem is referenced by:  xpscf  16226  xpsff1o  16228